Compare the formula for the volume of a cuboid with the formula for the area of a rectangle (length × width).
Answer: Area of a rectangle = l × w (two lengths, square units). Volume of a cuboid = l × w × h = (area of the rectangle) × height (three lengths, cubic units). The cuboid’s formula is the rectangle’s formula with one more length multiplied in.
Step-by-step solution
Idea: A rectangle is flat (two directions); a cuboid has a third direction, height. Each extra direction adds one more length to the product.
- Rectangle: area = length × width = lw. It counts unit squares, so the unit is cm2 or m2.½ mark
- Cuboid: volume = length × width × height = lwh. It counts unit cubes, so the unit is cm3 or m3.½ mark
- Comparison: lwh = (lw) × h. The volume is the area of the rectangle (the base) multiplied by the height. The cuboid is the ‘3D version’ of the rectangle.1 mark
Area of rectangle = lw (square units); volume of cuboid = lwh = (area of rectangle) × h (cubic units). The cuboid formula has one extra factor, the height.
Check: A rectangle 5 cm × 3 cm has area 15 cm2. A cuboid on it, 2 cm high, holds 15 × 2 = 30 cm3 = 5 × 3 × 2 ✓.
Answer to write in the exam
Area of rectangle = l × w (square units)
Volume of cuboid = l × w × h (cubic units)
∴ Volume of cuboid = (area of rectangular base) × height
Common mistakes that cost marks
- Writing volume in cm2 or area in cm3. Two lengths multiplied give square units; three give cubic units.
- Adding the height instead of multiplying: (lw + h) is not a volume.
How this can come in the exam
MCQ (1 mark)
A rectangle of area 24 cm2 is the base of a cuboid of height 5 cm. The volume of the cuboid is
- 29 cm3
- 120 cm2
- 120 cm3
- 48 cm3
Show answer
(C) 120 cm3
24 × 5 = 120, and volume is in cm3.
Try one yourself
A floor of a room is 6 m × 4 m and the room is 3 m high. Find the floor area and the volume of the room.
Show answer
Floor area = 24 m2; volume = 24 × 3 = 72 m3.
More questions like this
- Try to work out for yourself why this model explains the formula for the volume of a cuboid, i.e.,
volume = area of base × height.
Write the formula for the surface area and the volume of a cube. - Observe how these two formulas are special cases of the formulas 2(wl + hl + hw) and whl, when w = h = l.
- Two solid objects are made from the same material: Cube A has side 6 cm and Cuboid B has dimensions 9 cm × 6 cm × 4 cm. Compare their volumes and surface areas and determine which object has a greater surface area. How is this useful in a real-life situation?
- The volume of a cube is 64 cm3. What is its total surface area?
- How many small cubes with side 20 cm can be packed tight in a cubical box with side 2 m?